1 FV= Pv(1+r)t -1 r Fv = 5000( 1+12%) % 12% Suppose, that the financial manager decided to invest 5000 in the securities market at 12% to five years from now FV= PV(1+r)n = 5000(1+%12)5 = = 5000(1+%12)5 = Suppose that the financial manager decided to invest 5000$ at the end month for of 5 year from now to establish portfolio,(in the securities market at 12%)

2 We must adjust the accumulation factor = 5000 (1+(%12/12))5* = = 5000 (1+(%12/12))5* = (%12/12) (%12/12) PRESENT VALUE OF A CASH FLOW Suppose now that we will receive only $100 in two years time. What PV must we invest at 10%to give this cash flow two years hence? Pv(1+10%) ² = 100 Pv(1+10%) ² = 100 Pv = 100 = $82:64 (1+10%) ² (1+10%) ² Then pv = FV * 1 (1+r)t (1+r)t The rate E(R) used in the formula is the discount rate. PV tables is used to found accumulation factor

3 Notice :if the discount rate changes in each future period, the discount factor becomes : accumulation factor =. 1 ( (1+(r1))(1+(r2)) … (1+r3)) ( (1+(r1))(1+(r2)) … (1+r3)) or = or = (1+r)t. (1+r)t r PRESENT VALUE OF A PROJECT a firm invests in projects that have after-tax cash income each year for perhaps many future years One must multiply each such future cash flow by its discount factor before adding it to the rest. net present value (NPV) = PV for cash inflow – PV for cash outflow. The NPV tells us something very important, it gives us an estimate of how much value the project would add for shareholders

4 a leading American Internet products company considering investing $12 million in a new product. Management expects the product to generate incremental after-tax cash flows of $5 million each year for five years, starting in one year ’ s time. The company ’ s finance director estimates that the product ’ s asset beta equals 1.50 and that the Treasury bill rate will be 5% during the life of the product. From the Securities Market Line, Rm= 11% a leading American Internet products company considering investing $12 million in a new product. Management expects the product to generate incremental after-tax cash flows of $5 million each year for five years, starting in one year ’ s time. The company ’ s finance director estimates that the product ’ s asset beta equals 1.50 and that the Treasury bill rate will be 5% during the life of the product. From the Securities Market Line, Rm= 11% Required rate of return = rf + beta (rm- rf) = 5%+ 1.5(11%- 5%) Required rate of return = %14

5 Presen t value PV Factor amount s years NPV

6 PRECISION DISCOUNTING INTRA-YEAR DISCOUNTING For example, let us suppose that you want to discount a cash flow expected 500 days from now. We know that 500 days is 500/365= 1.37 years (rounded). Simply raise the discount factor to the 1.37 power: = 1 (1+r)1.37 or = 1 (1+r/365)500 ANNUITY FACTOR or = (1+r)t. (1+r)t r

7 equal annual payment on loans = Present value of loan Accumulation factor =1/r (1-1/(1+r)n or = (1+r)t. (1+r)t r EXAMPLE 2.13 E Book #35

8 THE LOAN BALANCE METHOD The loan balance method recognizes that an instalment is a combined payment of interest and part repayment of the loan. When the bank receives an instalment, it takes the interest owed on the previous loan balance and uses the remainder to reduce the balance owed. Loan balance Repay ment Intere st InstalmentYear ending

9 THE INTERNAL RATE OF RETURN (IRR) is the discount rate that forces the NPV of the project to zero

10 Internal Rate of Return Model The IRR determines the interest rate at which the NPV equals zero. If IRR > minimum desired rate of return, then NPV > 0 and accept the project. If IRR < minimum desired rate of return, then NPV < 0 and accept the project.

11Example: Original investment (cash outflow): = $ Useful life: five years. the following table show Annual income generated from investment Cash inflows Years Minimum desired rate of return: 15%. Determine IRR

12 1 – Assume discount rate = 10% yearsamount PV facto r present value

13 1 – Assume discount rate = 18% present value PV facto r amountyears

14 1 – Assume discount rate = 20% present value PV facto r amountyears IRR= %18+ %2*(882.2/1388.1) =%19.27, IRR > minimum desired rate of return %15 the should accept the project

15 THE VALUE OF GROWTH